Relationship between Generalized Orthogonality and Gâteaux Derivative
This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of <i>x</i> in the <i>y</i> direction is zero when the norm is Gâteaux differentiable in the <i&g...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-01-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/3/364 |
| Summary: | This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of <i>x</i> in the <i>y</i> direction is zero when the norm is Gâteaux differentiable in the <i>y</i> direction at <i>x</i> and <i>x</i> and <i>y</i> satisfy certain generalized orthogonality conditions. A case where <i>x</i> and <i>y</i> are approximately orthogonal is also analyzed and the value range of the Gâteaux derivative in this case is given. Moreover, two concepts are introduced: the angle between vectors in normed linear space and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>⊥</mo><mo>Δ</mo></msub></semantics></math></inline-formula> coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper. |
|---|---|
| ISSN: | 2227-7390 |